Testing Block Sphericity
of a Covariance Matrix
Prueba de Esfericidad por Bloques
para una Matriz de Covarianza
Liliam Carde˜
no (
lacero@matematicas.udea.edu.co
)
Daya K. Nagar (
nagar@matematicas.udea.edu.co
)
Departamento de Matem´aticas Universidad de Antioquia, Medell´ın
A. A. 1226, Colombia
Abstract
This article deals with the problem of testing the hypothesis thatq p-variate normal distributions are independent and that their covariance matrices are equal. The exact null distribution of the likelihood ratio statistic whenq= 2 is obtained using inverse Mellin transform and the definition of Meijer’sG-function. Results forp= 2, 3, 4 and 5 are given in computable series forms.
Key words and phrases: block sphericity, distribution, inverse Mellin transform, Meijer’sG-function, residue theorem.
Resumen
Este art´ıculo trata el problema de probar la hiptesis de que q dis-tribuciones normales p-variadas son independientes y con matrices de covarianza son iguales. La distribuci´on exacta nula del estad´ıstico de raz´on de verosimilitudes cuando q = 2 se obtiene usando la transfor-mada inversa de Mellin y la definicin de la G-funcin de Meijer. Los resultados parap = 2, 3, 4 y 5 se dan en forma de series calculables.
Palabras y frases clave: esfericidad por bloques, distribuci´on, trans-formada inversa de Mellin, funci´onGde Meijer, teorema del residuo.
1
Introduction
Suppose that the pq×1 random vector X has a multivariate normal distri-bution with mean vector µ and covariance matrix Σ and that X, µ and Σ are partitioned as
X=¡
X′1 X′2 · · · X′q¢′
, µ=¡
µ′
1 µ′2 · · · µ′q
¢′
and
Σ =
Σ11 Σ12 · · · Σ1q
Σ21 Σ22 · · · Σ2q
.. .
Σq1 Σq2 · · · Σqq
where Xi and µi arep×1 and Σij isp×p,i, j= 1, . . . , q. Consider testing
the null hypothesis H that the subvectorsX1,X2, . . . ,Xq are independent
and that the covariance matrices of these subvectors are equal, i.e.,
H : Σ =
∆ 0 · · · 0
0 ∆ · · · 0 ..
.
0 0 · · · ∆
against the alternative Ha thatH is not true. In H the common covariance matrix ∆ is unspecified. Olkin, while extending the circular symmetric model to the case where the symmetries are exhibited in blocks, definedHand called
it block sphericity hypothesis (see [13]). LetA be the sample sum of squares
and product matrix formed from a sample of size N =n+ 1. PartitionAas
A= (Aij) whereAij isp×p,i, j= 1, . . . , q. The likelihood ratio statistic for
testingH (see [17]) is given by
Λ = det(A)
1 2N
det³1qPq
i=1Aii
´12qN.
The null hypothesis is rejected if Λ≤Λ0 where Λ0 is determined by the null
distribution and level of significance. When p = 1, the hypothesis of block sphericity reduces to the Mauchly sphericity hypothesisH : Σ =σ2I
q which
[6, 7], Muirhead [10], Gupta [3], Nagar, Jain and Gupta [11] and Amey and Gupta [1]. The hypothesis of block sphericity is useful in multivariate repeated measures designs (see [17]). Since,A∼Wpq(n,Σ), thehth null moment of Λ can be obtained by integrating over Wishart density. We have
E(Λh) =q1
Theorem 3.3.22 of [5], we obtain
E
Substituting (1.3) in (1.2) and using (1.1) we obtain the hth null moment of
Λ as
distribution of −2 ln Λ is chi-square with 12p(q−1)(pq+p+ 1) d.f. The null distribution of ΛN2, in series involving Bernoulli polynomials, is available in
[?, CG]
In this article we will derive the exact null distribution of V = ΛN1 by
using inverse Mellin transform and residue theorem (see [14, 12]).
2
Exact density of
V
In order to obtain the exact density functionf(v) of V = ΛN1 we start with
the null moment expression. Substitutingq= 2 in (1.4) thehth moment ofV
is simplified as
E(Vh) = 2ph
2p
Y
j=1
½Γ[1 2h+
1
2(n−j+ 1)]
Γ[12(n−j+ 1)]
¾ p Y
j=1
½ Γ[
n−12(j−1)] Γ[h+n−12(j−1)]
¾
This will be rewritten in a slightly different form by using duplication formula for gamma function
Γ(2z) = 2
2z−1
√
π Γ(z)Γ
µ
z+1 2
¶
.
Then
E(Vh) =K(n, p)
p
Y
j=1
½
Γ(h+n−2p−1 + 2j) Γ[h+n−12(j−1)]
¾
(2.1)
where
K(n, p) =
p
Y
j=1
½ Γ[n
−12(j−1)]
Γ(n−2p−1 + 2j)
¾
. (2.2)
Now the density functionf(v) ofV = ΛN1 is obtained by taking inverse Mellin
transform ofE(Vh) as
f(v) = (2πι)−1Z
L
E(Vh)v−h−1dh,0< v <1, (2.3)
where ι =√−1 andL is a suitable contour. Substituting (2.1) in (2.3) and applying the transformationh+n−2p=t, one obtains
f(v) =K(n, p)vn−2p−1(2πι)−1Z
L1
p
Y
j=1
½
Γ(t+ 2j−1) Γ[t+ 2p−12(j−1)]
¾
v−tdt,
where L1 is the changed contour andK(n, p) is defined in (2.2). Now, using
the definition of Meijer’sG-function (see [8]), the density ofV is written as
f(v) =K(n, p)vn−2p−1Gp,0
p,p
·
v¯¯ ¯
2p−12(j−1), j= 1, . . . , p
2j−1, j= 1, . . . , p
¸
,0< v <1,
where Gp,0
p,p[ ] is the Meijer’sG-function.
Explicit expressions for the density of V for particular values of p will be obtained by evaluating the integral in (2.4) with the help of of residue theorem.
From (2.4), the density for p= 2 is obtained as
f(v) =K(n,2)vn−5(2πι)−1
Z
L1
Γ(t+ 1) (t+ 3)Γ(t+72)v
−tdt, 0< v <1. (2.5)
The integrand has simple poles at t=−r−1, r= 0,1,3,4, . . ., and a pole of order two att=−3. The residue att=−r−1,r6= 2, is
lim
t→−r−1
·
(t+ 1 +r)Γ(t+ 1) (t+ 3)Γ(t+72) v
−t
¸
= lim
t→−r−1
·
(t+ 1 +r)(t+ 1 +r−1)· · ·(t+ 1)Γ(t+ 1) (t+ 1 +r−1)· · ·(t+ 1)(t+ 3)Γ(t+7
2)
v−t
¸
= lim
t→−r−1
· Γ(
t+ 1 +r+ 1)
(t+ 1 +r−1)· · ·(t+ 1)(t+ 3)Γ(t+7 2)
v−t
¸
= 1
(−1)r+1r! (r−2)Γ(5 2−r)
vr+1.
Simplifying Γ(52 −r) in the above expression using the result Γ(β −j) =
(−1)jΓ(β)Γ(1−β)
Γ(1−β+j) we obtain the residue att=−r−1 as
−π1 Γ(r−
3 2)
r! (r−2)v
r+1, r
6
The residue att=−3 is derived as
where ψ( ) is the psi function (ver [9]). Now applying the residue theorem to the right hand side of (2.5), the density f(v) is obtained as using residue theorem, the density in this case is obtained (for 0< v <1) as
The integrand has simple poles att=−1,−2, poles of order two att=−1−r,
r= 2,3,7,8, . . ., and poles of order three att=−1−r,r= 4,5,6. Evaluating residues at these poles and using the residue theorem, we get the density for
The method employed above can also be used to derive the exact density for p≥6. However, because of the higher order of poles, the expressions for the density will involve generalized Riemann zeta function (see [9]).
Acknowledgment
This research was supported by the Comit´e para el Desarrollo de la Investi-gaci´on, Universidad de Antioquia research grant no. IN358CE.
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